The techniques developed in this paper apply to a wide range of other
geometric examples. More explicitly, we can use them in all cases
when local deformations admit
a simple DGLA description in terms of a resolution of the object to be
deformed, for instance, in the case of infinitesimal deformations of a
singular variety. Namely, let \(X\)  be a singular variety, \({\mathcal {O}}_X\)  the sheaf of regular function of \(X\)  and \({\mathcal {R}}^\cdot \rightarrow {\mathcal {O}}_X\)  its standard free resolution [1]}. Then, the deformation functor of infinitesimal
deformations of \(X\)  is isomorphic to \( H_{\rm Ho}^1(X; \exp {{\mathcal {D}}er}^*({\mathcal {R}}^\cdot ) )\) ; see [2]} for details. From
this, we also recover the classical result that the tangent space to
deformations of \(X\)  is \(\operatorname{Ext}^1({\mathbb {L}}_X,{\mathcal {O}}_X)\) , and
that obstructions are contained in \(\operatorname{Ext}^2({\mathbb {L}}_X,{\mathcal {O}}_X)\) , where \({\mathbb {L}}_X\)  is the cotangent complex of \(X\) .
